Multi-dimensional $\rho$-almost periodic type functions and applications

Abstract

In this paper, we analyze various classes of multi-dimensional $\rho$-almost periodic type functions $F : I \times X \rightarrow Y$ and multi-dimensional $(\omega,\rho)$-almost periodic type functions $F : I \times X \rightarrow Y,$ where $n\in {\mathbb N},$ $\emptyset \neq I \subseteq {\mathbb R}^{n},$ $X$ and $Y$ are complex Banach spaces and $\rho$ is a binary relation on $Y.$ The proposed notion is new even in the one-dimensional setting, for the functions of the form $F : I \rightarrow Y.$ The main structural properties and characterizations for the introduced classes of functions are presented. We provide certain applications of our theoretical results to the abstract Volterra integro-differential equations, as well.